First hyperbolic times for intermittent maps with unbounded derivative
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Bibliographic record
Abstract
We establish some statistical properties of the hyperbolic times for a class of non-uniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric baker’s map-type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last 30 years. The statistics of these maps are controlled by the order of tangency (linked to a single parameter α, where 0 < α < ∞) that a certain ‘cut function’ makes with the boundary of the square. Previously, a direct Young tower construction has been used to obtain optimal correlation decay rates of O(n−1/α) for Hölder observables and all values of the parameter α. A central limit theorem (CLT) is obtained when 0 < α < 1.The asymptotics of a natural hyperbolic time for this family of maps are analysed via the same Young tower. By using a large deviations result of Melbourne and Nicol, we prove that the first hyperbolic time is integrable if and only if the parameter satisfies 0 < α < 1. Furthermore, within this restricted range of parameters, concentration inequalities recently established by Chazottes and Gouëzel imply sharp O(n−1/α) bounds on the tail distribution of first hyperbolic times. As shown by Alves, Viana, and others, knowledge of the tail distribution of the hyperbolic times leads to upper bounds on the rate of decay of correlations and derivation of a CLT. Comparing to the results obtained directly for this family of maps, the latter estimates via hyperbolic times are suboptimal, even over the restricted range of parameters 0 < α < 1.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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